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JEE Main20199 Apr 2019Morning ShiftMathematicsApplication of DerivativesActual

If f ( x ) is a non-zero polynomial of degree four, having local extreme points at x = – 1 , 0 , 1 ; then the set S = x ∈ R : f x = f 0 contains exactly

Options

  1. ATwo irrational and two rational numbers
  2. BFour rational numbers
  3. CTwo irrational and one rational number
  4. DFour irrational numbers

Correct answer

C. Two irrational and one rational number

Step-by-step solution

f ( x ) has extreme points at - 1 ,   0 ,   1 ⇒ f ' x = 0   at x =   – 1 ,   0 ,   1 ⇒ f ' x = a x 2 - 1 x     ∵ f   x is a polynomial ⇒ f x = ∫ a x 3 - x d x = a x 4 4 - x 2 2 +   C Now the equation, f x = f ( 0 ) ⇒ a x 4 4 - x 2 2 +   C = C ⇒   x 4 4 - x 2 2 = 0   ⇒ x 2 x 2 - 2 = 0 ⇒ x = 0 ,   ± 2 ⇒ S =   0 ,   2 ,   - 2 . Hence, the set contains two irrat

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